On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity

Abstract

The article is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second order elliptic problems by Chipot and Rougirel in [3], where the force functions are considered on the cross section of domains, we prove the non-local counterpart of their result.Furthermore, recently in [15] Yeressian established a weighted estimate for solutions of nonlocal Dirichlet problems which exhibit the asymptotic behavior. The case when s = 1/2 was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this article, we extend this result to each order between 0 and 1.

Study of fractional Poincaré inequalities on unbounded domains

“…By definition P 1 n,s (Ω) ≤ P 2 n,s (Ω), so we also have that P 1 n,s (Ω) = 0. The following lemma relates the constants C n,s for different values of the dimension n. It is a generalization of the Lemma 3.1 in [10]. We will use these algebraic relations several times.…”

“…denotes the principal value and the above integral is defined for u ∈ C 2 c (R n ). We refer to [10], [20], [31], [37] and [36] for related works concerning the fractional Laplace operator. The constant C n,s is explicitly given by…”

On some variational problems set on domains tending to infinity

“…Apart from second order elliptic equations [10] already mentioned, this includes eigenvalue problems, parabolic problems, variational inequalities, Stokes problem, hyperbolic problems and many others. We refer to [3,5,6,7,8,9,11,13,14,19,20] and the references there for the literature available in this direction.…”

Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded

“…In the present paper we deal with the behaviour of the Dirichlet fractional Laplacian on varying (n + k)-dimensional domains which become unbounded in the last k-directions. For related results we cite [8,23], where k = 1 is assumed, and the recent paper [7], where the regional (or restricted) Dirichlet fractional Laplace operator is considered.…”

“…By elementary variational arguments, problem (1.2) admits a unique solution u ℓ = u ℓ (x, t) ∈ H s (Ω n+k ℓ ). A description of the asymptotic behaviour of u ℓ in case k = 1, s ∈ ( 1 2 , 1), has been given in [8,Theorem 1.2]. It turns out that…”